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Theorem psubclsubN 39445
Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclsub.s 𝑆 = (PSubSp‘𝐾)
psubclsub.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
psubclsubN ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋𝑆)

Proof of Theorem psubclsubN
StepHypRef Expression
1 eqid 2728 . . 3 (⊥𝑃𝐾) = (⊥𝑃𝐾)
2 psubclsub.c . . 3 𝐶 = (PSubCl‘𝐾)
31, 2psubcli2N 39444 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)
4 eqid 2728 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
54, 1, 2psubcliN 39443 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐶) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋))
65simpld 493 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋 ⊆ (Atoms‘𝐾))
7 psubclsub.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
84, 7, 1polsubN 39412 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘𝑋) ∈ 𝑆)
96, 8syldan 589 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘𝑋) ∈ 𝑆)
104, 7psubssat 39259 . . . 4 ((𝐾 ∈ HL ∧ ((⊥𝑃𝐾)‘𝑋) ∈ 𝑆) → ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
119, 10syldan 589 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
124, 7, 1polsubN 39412 . . 3 ((𝐾 ∈ HL ∧ ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝑆)
1311, 12syldan 589 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝑆)
143, 13eqeltrrd 2830 1 ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wss 3949  cfv 6553  Atomscatm 38767  HLchlt 38854  PSubSpcpsubsp 39001  𝑃cpolN 39407  PSubClcpscN 39439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-proset 18294  df-poset 18312  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p1 18425  df-lat 18431  df-clat 18498  df-oposet 38680  df-ol 38682  df-oml 38683  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-psubsp 39008  df-pmap 39009  df-polarityN 39408  df-psubclN 39440
This theorem is referenced by:  pclfinclN  39455
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